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Prime Factors

In this section we learn about prime factors and how to find a whole number's prime factors.

A prime factor of a number is a prime number, which is also a factor of that number.

Remember that a prime number \(p\) is a whole number that has exactly two factors (a.k.a divisors):

  • itself: \(p\), and
  • 1
For example, \(7\) is a prime number as it only has two factors: \(1\) and \(7\).

The first few prime numbers are: \[2,3,5,7,11,13,17,\dots\] Every whole number greater than, or equal to, \(2\) has at least one prime factor.

Definition - Prime Factor

Given a whole number \(m\), a prime factor \(p\) of \(m\) is a factor of \(m\) which is also a prime number.
For instance one of the prime factors of \(12\) is \(3\), as \(3\) is a factor of \(12\) and \(3\) is a prime number; indeed we can write: \[12 = 3\times 4\] so \(3\) is a factor of \(12\) and since \(3\) is a prime number it is a prime factor of \(12\).

Method - Listing a Number's Prime Factors

Given a whole number \(m\), we can list all of its prime factors, \(p_1,p_2,p_3,\dots \), using the following two steps:

  • Step 1: Write a list of the first few prime numbers: \(2, 3, 5, 7, 11, 13, 17, 19, \dots \) .

  • Step 2: make the list of prime factors using all the factors of \(m\) you see in the list of prime numbers, written in step 1.
This two-step method is further explained in the following tutorial:

Tutorial: Listing a Number's Prime Factors

In the following tutorial we learn how to find the prime factors of a whole number.

Exercise 1

List all of the prime factors of each of the following whole numbers

  1. \(6\)

  2. \(30\)

  3. \(12\)

  4. \(42\)

  5. \(98\)

Answers Without Working

  1. The prime factors of \(6\) are \(2\) and \(3\).

  2. The prime factors of \(30\) are \(2\), \(3\), and \(5\).

  3. The prime factors of \(12\) are \(2\) and \(3\).

  4. The prime factors of \(42\) are \(2\), \(3\) and \(7\).

  5. The prime factors of \(98\) are \(2\) and \(7\).

Solution With Working

  1. To find the prime factors of \(6\), we follow our two-step method:

    • Step 1: list the first few prime numbers: \[2,3,5,7,11,13, \dots \]

    • Step 2: find all of the factors of \(6\) inside the list, made in step 1:

      Looking at the list, the only factors of \(6\) are: \(2\) and \(3\).
    • The prime factors of \(6\) are therefore \(2\) and \(3\).

  2. To find the prime factors of \(30\), we follow our two-step method:

    • Step 1: list the first few prime numbers: \[2,3,5,7,11,13, \dots \]

    • Step 2: find all of the factors of \(30\) that are in the list, written in step 1:

      Looking at the list, the only factors of \(30\) are: \(2\), \(3\), and \(5\).
    The prime factors of \(30\) are therefore \(2\), \(3\), and \(5\).

  3. To find the prime factors of \(12\), we follow our two-step method:

    • Step 1: list the first few prime numbers: \[2,3,5,7,11,13, \dots \]

    • Step 2: find all of the factors of \(12\) contained in the list, written in step 1:

      Looking at the list, the only factors of \(12\) are: \(2\) and \(3\).
    The prime factors of \(12\) are therefore \(2\) and \(3\).

  4. To find the prime factors of \(42\), we follow our two-step method!

    • Step 1: list the first few prime numbers: \[2,3,5,7,11,13, \dots \]

    • Step 2: find all of the factors of \(42\) contained in the list, written in step 1:

      Looking at the list, the only factors of \(42\) are: \(2\), \(3\) and \(7\).
    • The prime factors of \(42\) are \(2\), \(3\) and \(7\).

  5. To find the prime factors of \(98\), we follow our two-step method:

    • Step 1: list the first few prime numbers: \[2,3,5,7,11,13, \dots \]

    • Step 2: find all of the factors of \(98\) contained in the list, written in step 1:

      Looking at the list, the only factors of \(98\) are: \(2\) and \(7\).
    • The prime factors of \(98\) are \(2\) and \(7\).

Exercise 2

State whether each of the statements that follow is true or false.

  1. \(4\) is a prime factor of \(12\).

  2. \(5\) is a prime factor of \(20\).

  3. \(11\) is a prime factor of \(21\).

  4. \(2\) and \(7\) are both prime factors of \(28\).

  5. \(9\) is a prime factor of \(27\).

Answers Without Working

  1. false

  2. true

  3. false

  4. true

  5. false

Solution with Working

  1. false: \(4\) is a factor of \(12\) but it isn't a prime number so it isn't one of \(12\)'s prime factor's.

  2. true: \(5\) is a prime factor of \(20\) since \(5\) is a prime number and \(5\) is a factor of \(20\), indeed: \(20 = 5\times 4\).

  3. false: \(11\) is a prime number but it isn't a factor of \(21\) so it isn't one of its prime factors.

  4. true: both \(2\) and \(7\) are prime numbers and they are both factors of \(28\). Indeed:
    • \(28 = 2\times 14\), and

    • \(28 = 7\times 4\).

  5. false: \(9\) is a factor of \(27\) but \(9\) isn't a prime number so it isn't a prime factor.

Prime Factorization

All whole numbers can be written as a product of its prime factors.

Method

The method we learn here is based-on the fundamental theorem of arithmetic. In essence, it states:

All whole numbers greater than, or equal to \(2\), can be written as a product of its prime factors.

Tutorial: Writing a Number as the Product of its Prime Factors

In the following tutorial we learn how to write a number as a product of prime factors, watch it now:

Exercise 3

Write each of the following whole numbers as its product of prime factors:

  1. \(36\)

  2. \(90\)

  3. \(196\)

  4. \(900\)

  5. \(384\)

  6. \(3600\)

  7. \(210\)

  8. \(540\)

Solution Without Working

  1. \(36 = 2^2\times 3^2\)

  2. \(90 = 2\times 3^2\times 5\)

  3. \(196 = 2^2\times 7^2\)

  4. \(900 = 2^2\times 3^2 \times 5^2\)

  5. \(384 = 2^7\times 3\)

  6. \(7200 = 2^4 \times 3^2 \times 5^2\)

  7. \(210 = 2\times 3\times 5\times 7\)

  8. \(540 = 2^2 \times 3^2 \times 5\)